Wolfram Cloud Document

ClearAll["Global`*"];

NN=1;M=1;



m

=NN*(M+1);α=0.9;q=0.7;T=1;Unprotect[Power];Power[0|0.,0|0.]=1;Protect[Power];

g[x_]:=g[x]=x+

2

q+1

x

Gamma[q+2]

F[m_][x_]:=F[m][x]=

IntegerPart

m

2



∑

k=0

Binomial[m-k,k]*

m-2*k

x

f[n_,m_][x_]:=f[n,m][x]=PiecewiseF[m][NN*

α

x

-n+1],

1

α

n-1

NN

≤x<

1

α

n

NN

,0

θ1[x_]:=θ1[x]=FlattenTablePiecewiseF[m][NN*

α

x

-n+1],

1

α

n-1

NN

≤x<

1

α

n

NN

,0,{n,1,NN},{m,0,M}

U[n_,m_][β_,x_]:=U[n,m][β,x]=

IntegerPart

m

2



∑

k=0

m-2*k

∑

r=0

Binomial[m-k,k]*Binomial[m-2*k,r]*

r

NN

*

m-2*k-r

(1-n)

*

α*r+β

x

*

Gamma[α*r+1]

Gamma[β+α*r+1]

*1-BetaRegularized

1

α

n-1

NN

*

1

x

,α*r+1,β//Expand

V[n_,m_][β_,x_]:=V[n,m][β,x]=

IntegerPart

m

2



∑

k=0

m-2*k

∑

r=0

Binomial[m-k,k]*Binomial[m-2*k,r]*

r

NN

*

m-2*k-r

(1-n)

*

α*r+β

x

*

Gamma[α*r+1]

Gamma[β+α*r+1]

*1-BetaRegularized

1

α

n

NN

*

1

x

,α*r+1,β//Expand

θ1

[β_][x_]:=

θ1

[β][x]=FlattenTablePiecewise0.,-Infinity<x<

1

α

n-1

NN

,U[n,m][β,x],

1

α

n-1

NN

≤x<

1

α

n

NN

,U[n,m][β,x]-V[n,m][β,x],

1

α

n

NN

≤x<Infinity,{n,1,NN},{m,0,M}

A=Arrayc,



m



{c[1],c[2]}

subdivision=NTablex[l]=

1

q

l+1

2*



m

,l,0,



m

-1

{0.138011,0.371499}

eqn1[x_]=PiecewiseExpandA.θ1[x]+

2

A.

θ1

[q][x]

-g[x]

-0.419102(2.38605x+1. 3.4 x )	x<0.
-0.419102(2.38605x+1. 3.4 x -2.38605c[1]-2.89 1.4 x 2 c[1] -2.38605 0.9 x c[2]-3.53319 2.3 x c[1]c[2]-1.07988 3.2 x 2 c[2] )	0.≤x<1.
-0.4191022.38605x+1. 3.4 x -2.89 1.4 x 2 c[1] +5.78 0.7 1- 1. x 1.4 x 2 c[1] -2.89 1.4 1- 1. x 1.4 x 2 c[1] -3.53319 2.3 x BetaRegularized 1. x ,1.9,0.7c[1]c[2]+3.53319 0.7 1- 1. x 2.3 x BetaRegularized 1. x ,1.9,0.7c[1]c[2]-1.07988 3.2 x 2 BetaRegularized 1. x ,1.9,0.7 2 c[2]	True

eqn=Join[Map[eqn1,subdivision]]

{-0.419102(0.330493-2.38605c[1]-0.180625

2

c[1]

-0.401424c[2]-0.0371509c[1]c[2]-0.0019103

2

c[2]

),-0.419102(0.920919-2.38605c[1]-0.7225

2

c[1]

-0.978682c[2]-0.3623c[1]c[2]-0.045419

2

c[2]

)}

sol=FindRooteqn0,TransposeJoin[A],ConstantArray0,



m



{c[1]-0.024306,c[2]0.965247}

sol1[x_]=PiecewiseExpand[A.

θ1

[q][x]/.sol]

0.	x<0.
-0.0267499(1. 0.7 x -24.2752 1.6 x )	0.≤x<1.
0.0267499-1. 0.7 x +1. 0.7 1- 1. x 0.7 x +24.2752 1.6 x BetaRegularized 1. x ,1.9,0.7	True

p1=ListPlot[Table[{x,sol1[x]},{x,0,T,0.1}],PlotRangeAll]

p2=Plot

q+1

x

Gamma[q+2]

,{x,0,1},PlotStyleRed

Show[p1,p2]